The Nyquist-Shannon theorem (the sampling theorem) states that if a function x(t) contains no frequencies greater than B hertz, it is completely characterized by measuring its amplitudes at a series of points spaced 1/(2B) seconds apart. It further states that to reconstruct the original signal one must use the Whittaker-Shannon interpolation theorem, computing the sum of the infinite series of normalized sinc functions for each sample. (Well,

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$x(t)=\sum_{-\infty}^{\infty}x[n]\cdot{\rm sinc}(\frac{t-nT}{T})$

where n is the sample number, t is the sample time, T is 1/(sampling rate), and sinc(x) is the normalized sinc function.)

I've never (in my somewhat limited experience) seen an audio DAC that does that, yet they seem to have pretty reasonable output. How accurate are the standard approximations, really? What is the interpolation error? How does one characterize it?

The formula of Whittaker-Shannon contains infinite sums. Practical DACs typically evaluate this sum over 20–100 samples which is a good approximation: it allows for nearly perfect reconstruction of signals up to 0.45•Fs.